Good lattice points, discrepancy, and numerical integration
Good lattice points, discrepancy, and numerical integration
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DOI：
10.1007/bf02415091
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发表时间：
1966-12
影响因子：
1
通讯作者：
S. Zaremba
中科院分区：
文献类型：
--
作者：
S. Zaremba
基于丢番图近似的方法导致了有理向量的简单且容易的构造，有理向量的倍数减少模1，形成有限的点序列，在单位正方形上具有一定的均分性质。给定该平方上有界变化的函数，建议计算该序列的各点处的值的平均值可以是数值积分的实用方法。获得的错误矿石的精确范围。在一般情况下，这些是点数的倒数和对数的乘积的量级；在函数满足规定的规律性和周期性条件的情况下，它们的数量级为点数的对数除以该数的适当高的幂。偶然获得了一些众所周知的均分结果的轻微锐化。
Methods based on Diophantine approximations lead to a simple and easy construction of rational vectors the multiple of which, reduced modulo 1, form finite sequences of points with certain properties of equipartition over the unit square. Given a function of bounded variation over this square, it is suggested that computing the average of its values at the points of such a sequence can be a practical method of numerical integration. Precise bounds for the error ore obtained. In the general case, these are of the order of the product of the reciprocal and of the logarithm of the number of points; in the case of a function satisfying stated conditions of regularity and periodicity, they are of the order of the logarithm of the number of points divided by an appropriately high power of this number. A slight sharpening of some well-known results on equipartiton is obtained incidentally.